1. Field of the Invention
The invention relates to a method for constructing bodies which, while rotating codirectionally at the same rotational speed about axes arranged in parallel, constantly touch one another at at least one point.
2. Description of Related Art
Let two circles be considered which are arranged next to one another on two parallel axes, as illustrated diagrammatically in FIG. 1. It is generally known that, during codirectional rotation, the circles brush against one another in such a way that, during rotation, they constantly touch one another at a point which lies between the centres of rotation of the circles.
It is known, furthermore, that, in addition to circles, there are further geometric figures which constantly touch one another at a point during codirectional rotation. One example is shown in FIG. 2. These figures constantly touch one another at one point when they are rotating codirectionally at the same rotational speed.
The two-dimensional geometric figures shown in FIGS. 1 and 2 can be continued in various ways into the third dimension. A simple possibility is, for example, the linear continuation of the figures in the direction of the axes of rotation, so as to give rise to disc-shaped or rod-shaped bodies which, during codirectional rotation, brush against one another along a line between the centres of rotation which runs parallel to the axes of rotation.
A further possibility is, for example, to continue the geometric figures along the axes of rotation in a screw-shaped manner, so as to give rise to screw-shaped bodies which, during codirectional rotation, touch one another along a curve between the bodies.
Such bodies which constantly touch one another at at least one point when they rotate codirectionally at the same rotational speed about axes arranged in parallel are important particularly in extruder technology where they are used as codirectionally rotating screw-type extruders, for example for the processing of viscous masses or for mixing purposes. Such synchronous two-shaft and multishaft extruders are known to a person skilled in the art from the patent and specialized literature. The following publication [1] may be mentioned here as an example: K. Kohlgrüber: “Der gleichläufige Doppelschneckenextruder”, [“The synchronous double screw extruder”], Hanser Verlag, 2007. In screw-type extruders, the property whereby adjacent screws brush against one another in pairs when they rotate codirectionally has the advantage that they scrape off one another and therefore clean off one another.
For selected bodies which, when they rotate codirectionally at the same rotational speed about axes arranged in parallel, constantly touch one another at at least one point, there are regulations for their construction.
Thus, for example, it is known from the literature for screw-type extruders (see, for example, [1] pages 96 to 98] that a screw element of the “Erdmenger” type with a cross-sectional profile, as in FIG. 2 of the present application, can be assembled from arcs of a circle.
It is well known, however, what criteria have to be fulfilled in general so that two bodies rotating codirectionally about two axes arranged in parallel touch one another constantly at at least one point.
It is known (see, for example, [2]: Booy “Geometry of fully wiped twin-screw equipment”, Polymer Engineering and Science 18 (1978) 12, pages 973-984), that the codirectional rotation of two touching bodies about their fixed axes is kinematically equivalent to the “translation without rotation” of one body about the other body which is then stationary. This particular feature can be used for generating in steps geometric figures which constantly touch one another at one point during codirectional rotation. In the consideration, the first figure (the “generated” figure) is stationary and the second figure (the “generating” figure) is displaced about the first in translational motion on an arc of a circle. Part of the profile of the second figure can then be stipulated and it is possible to investigate which profile is thereby generated on the first figure. The generated figure is as it were “cut out” by the generating figure.
However, no general method is known as to how that part of the second figure which is stipulated can itself be generated. In [2], one possible approach is described as to how the profile segment which can be the starting point and from which the remaining profile is generated can be generated. However, this approach is highly complicated in mathematical terms and, above all, is not generally valid, that is to say only those profiles which can be described by the mathematical functions specified in [2] can be generated.